4.2 The ’abridging Molodensky’ datum parametrization method

The easiest way to define the connection between two datums is to define the vector connecting their geometric centers (Fig. 17). This vector should be given by the components in the geocentric Cartesian coordinate system, described in the Chapter 2, expressed in meters. Obviously, if both analyzed datum are of absolute dislocations, this vector is the null vector, with the components of (0 m; 0 m; 0 m). It should be noted that the international literature often and erroneously called this method as Molodensky- or Molodensky-Badekas-type parametrization, albeit they are indeed more complex ones.

Shift of an ellipsoid to a new position

Fig. 17. The abridged Molodensky transformation is a simple shift between two datum ellipsoids, expressed by the three components of the shift vector.

So, the three parameters of the abridging Molodesky datum description are the metric distances of dX, dY and dZ, describing the spatial locations of the geometric centers of the datum locations from each other. If one of these datums is the WGS84, these dX, dY and dZ parameters give the location of the local datum with respect to the mass center of the Earth. If the coordinates of a basepoint are known on a Datum ’1’, the geocentric coordinates on the Datum ’2’ are the following:

(4.2.1)

The angular difference between the coordinates on the starting and the goal datums can be also expressed without to convert to geocentric coordinates and vice versa:

(4.2.2)

(4.2.3)

(4.2.4)

where is the curvature in the prime meridian; is the curvature in the prime vertical; ΔΦ” and ΔΛ” are the latitude and logitude differences between the coordinates of the two datums in arc second; Δh is the difference between the ellipsoidal heights; a and f are the semi-major axis and the flattening of the starting datum; while da and df are the differences of them between the starting and goal datums. If the ellipsoidal heights are not given, they can be estimated from leveled heights using geoid models, or we can simply omit the Equation (4.2.4) at the calculation.

As it was mentioned, the GIS packages describe the datums by parameters between them and the WGS84 special datum, thus handling the problem that a datum cannot be this way parametrized alone, just the difference between it and another datum. If we have two different datums (not the WGS84) and we know the parameters of the transformation from them to the WGS84, the abridging Molodensky parameters between the two datums can be given because of the linearity. Let the transformation A is the one between the first datum and the WGS84 and the transformation B is the one from the second datum to it. The C shows the direct transformation from the first and second datums. The parameters of this C are (commutation):

(4.2.5)

These parameters are not depending on the ellipsoids used for the different datums. For example, the datum shift parameters from the Austrian MGI datum to the WGS84 are dX=+592 m; dY=+80 m; dZ=+460 m. The same parameter set between the German DHDN77 system and the WGS84 are dX=+631 m; dY=+23 m; dZ=+451 m. Thus, the direct transformation parameters from the MGI to the DHDN77 are dX=–39 m; dY=+57 m; dZ=+9 m.

In the literature, we often find different number triplets as parameters of a transformation from a specific datum and the WGS84. Albeit it is obviously an error in spatial context, the transformation error in the horizontal coordinates (latitude and longitude) is not necessarily significant at them. Using different triplets as abridging Molodensky parameters for a datum, as it is shown below, there is always one point on the ellipsoid, where the two different parameter set result the same horizontal shift. The main question is, whether this point falls to the extents of the valid territory of the datum (the geodetic network), if possible, near to its center/fundamental point, or not. If yes, both parameter sets can be used and we can compute the vertical difference of the two datums at that point. Usually, the difference is because of the neglecting of the geoid undulation value.

Let r1 the position vector from the center of the WGS84 to the geometric center of the Datum version 1 and r2 is the similar one to the center of the Datum version 2. Making the difference of these position vectors in the space:

rdiff = r1-r2

(4.2.6)

Now, let’s check that this vector shows to which point of the reference surface:

(4.2.7)

(4.2.8)

while the length of the difference vector (the spatial difference) in meters is

(4.2.9)

If the point (φrr) is in the area of the used triangulation network, possibly near to the fundamental point, both versions can be used. As I mentioned above, in this case, the length of rdiff is usually around the geoid undulation value between the local datum and the WGS84 at the point (φrr). If this point falls to a distant position on the Earth’s surface, one of the parameter sets is erroneous.